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+ N ] =x,[n + N ]+ x 2 [ n + N ] = x , [ n + mN,] +x,[n + kN2]
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= kN2 = N
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Thus, we must have
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Since we can always find integers m and k to satisfy Eq. (1.861, it follows that the sum of two periodic sequences is also periodic and its fundamental period is the least common multiple of N, and N,.
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1.16. Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period. 2T T ( a ) x ( t ) = cos ( b ) x(t)=sinpt
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( c ) x(t)=cos-I
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(el x ( t ) = sin2t ( g ) x[n] = ej("/4)"
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(dl x(t)=cost+sinfit ( f ) X(t) = eiI(r/2)f- 11 ( h ) x[n]=cosfn
x[n] = cos -n 3
+ sin -n 4
( j ) x[n] = cos2 -n
x(t) is periodic with fundamental period T , = 27r/w0
= 27r.
x(r) is periodic with fundamental period TO= 27r/o,,
lr lr
= 3.
( c ) x ( t ) = cos --I
+ sin -t = x , ( t ) +x2(t) 3 4 where x,(t) = cos(7r/3)r = cos w,t is periodic with T, = 27r/w, = 6 and x2(t) = s i n ( ~ / 4 ) t= sin w2t is periodic with T2 = 21r/w2 = 8. Since T,/T, = = is a rational number, x(t) is periodic with fundamental period To = 4T, = 3T2 = 24.
CHAP. 11
SIGNALS AND SYSTEMS
( d l x(t) = cos r + sin f i r =x,(r) +x2(r) where x,(t) = cos r = cos o , t is periodic with TI = 27r/01 = 27r and x2(t) = sin f i t = sin w2t is periodic with T2 = 27r/02 = fir. Since T,/T2 = fi is an irrational number, x(t) is nonperiodic. (e) Using the trigonometric identity sin2 0 = t(l - cos 201, we can write
where x,(t) = $ is a dc signal with an arbitrary period and x2(t) = - $ cos2r = - I cos 0 2 t is periodic with T2 = 2n/w2 = 7. Thus, x(t) is periodic with fundamental period To = T.
( f ) x(t) = ejt(r/2)r11 = e - j e j ( r / 2 ) r
= -I
'ej w d
7T ,I Wo = r
x(t) is periodic with fundamental period To = 27r/w0
= 4.
Since R0/27r = $ is a rational number, x[nl is periodic, and by Eq. (1.55) the fundamental period is No = 8. x[n] = cos f n = cos n o n --,R o = $ Since n0/27r = 1 / 8 ~ not a rational number, x[n] is nonperiodic. is x[n] = cos -n 3 where
+ sin -n 4
= x,[n]
+ x2[n1
x2[n] = sin -n = cos f12n +0, = 4 4 Since R , / 2 ~ r= (= rational number), xl[n] is periodic with fundamental period N, = 6, and since R2/27r = $ ( = rational number), x2[n] is periodic with fundamental period N2 = 8. Thus, from the result of Prob. 1.15, x[n] is periodic and its fundamental period is given by the least common multiple of 6 and 8, that is, No = 24. Using the trigonometric identity cos28 = i ( l x[n]
= cost
+ cos28), we can write
7 r =x,[n] +x2[n] 4
T -n 8
- + - cos -n
where x,[n] = $ = $(l)" is periodic with fundamental period Nl = 1 and x2[n] = 1 cos(a/4)n = cos R 2 n --, 2 = ~ / 4 .Since R2/27r = ( = rational number), x2[n] is Q periodic with fundamental period N2 = 8. Thus, x[n] is periodic with fundamental period No = 8 (the least common multiple of N, and N,).
1.17. Show that if x ( t
+ T ) = x ( t ) , then
for any real a,p, and a.
SIGNALS AND SYSTEMS
[CHAP. 1
If x(t and
+ T ) = x ( t ) , then letting t = 7 - T , we
have
X ( T - T + T ) = x ( r )= x ( T - T )
Next, the right-hand side of Eq. ( 1 . 8 8 )can be written as
By E q . ( 1 . 8 7 )we have
Thus.
x ( t ) dt
1.18. Show that if x ( t ) is periodic with fundamental period To, then the normalized average power P of x ( t ) defined by Eq. ( 1 . 1 5 ) is the same as the average power of x ( 0 over any interval of length T , , that is,
By Eq. ( 1.15)
P = lim - /T'2 1 x ( t ) 1' dt T-.r: T - 7 . / 2
Allowing the limit to be taken in a manner such that T is an integral multiple of the fundamental period, T = kT,, the total normalized energy content of x ( t ) over an interval of length T is k times the normalized energy content over one period. Then
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